Gallium oxide substrate and method of manufacturing gallium oxide substrate

ABSTRACT

A gallium oxide substrate includes first and second main surfaces. When measured data z 0 (r,θ) of height differences of points (r,θ,z) on the first main surface from a least square plane of the first main surface are approximated by a function z(r,θ)=Σa nm z nm (r,θ), a ratio of a first maximum height difference of a component of z(r,θ) obtained by summing terms a nm z nm (r,θ) with an index j of 4, 9, 16, 25, 36, 49, 64, and 81, when the second main surface is placed facing a horizontal flat surface, to a diameter of the first main surface is 0.39×10 −4  or less, and a ratio of a second maximum height difference of a component of z(r,θ) obtained by summing terms a nm z nm (r,θ) with j of from 4 to 81, when an entire surface of the second main surface is adsorbed to a flat chuck surface, to the diameter is 0.59×10 −4  or less.

CROSS-REFERENCE TO RELATED APPLICATION

The present application is a continuation application of InternationalApplication No. PCT/JP2020/011995, filed Mar. 18, 2020, which claimspriority to Japanese Patent Application No. 2019-073548 filed Apr. 8,2019. The contents of these applications are incorporated herein byreference in their entirety.

BACKGROUND OF THE INVENTION 1. Field of the Invention

The present disclosure relates to gallium oxide substrates and methodsof manufacturing gallium oxide substrates.

2. Description of the Related Art

Recently, compound semiconductor substrates have been used instead ofsilicon semiconductor substrates. Compound semiconductors include, forexample, silicon carbide, gallium nitride, and gallium oxide. Compoundsemiconductors are excellent in large band gaps compared with siliconsemiconductors. Compound semiconductor substrates are polished, andepitaxial films are formed on polished surfaces.

Japanese Unexamined Patent Application Publication No. 2016-13932discloses a method of manufacturing a gallium oxide substrate. Themethod includes polishing only one side of the gallium oxide substrateusing a slurry containing colloidal silica. The subject of JapaneseUnexamined Patent Application Publication No. 2016-13932 is to improve ashaping property of the gallium oxide substrate in which the crystalsystem is a monoclinic system having poor symmetry and strong cleavingproperty.

SUMMARY OF INVENTION Problem to be Solved by the Invention

Typically, a single-sided polishing device includes a lower surfaceplate, an upper surface plate, and a nozzle. The lower surface plate isarranged horizontally and a polishing pad is attached to an uppersurface of the lower surface plate. The upper surface plate is arrangedhorizontally and the gallium oxide substrate is fixed to a lower surfaceof the upper surface plate. The gallium oxide substrate has a first mainsurface and a second main surface opposite to the first main surface.The upper surface plate holds the gallium oxide substrate horizontallyand presses the first main surface of the gallium oxide substrateagainst the polishing pad. The lower surface plate is rotated around arotational center line orthogonal to the lower surface plate. The uppersurface plate rotates passively with the rotation of the lower surfaceplate. The nozzle supplies a polishing slurry from above to thepolishing pad. The polishing slurry is supplied between the galliumoxide substrate and the polishing pad. The first main surface of thegallium oxide substrate is flatly polished with the polishing slurry.Because the second main surface of the gallium oxide substrate is fixedto the lower surface of the upper surface plate, irregularities of thelower surface of the upper surface plate are transferred to the secondmain surface.

Because the single-sided polishing device polishes only the first mainsurface, after the polishing, a residual stress of the first mainsurface is different from the residual stress of the second mainsurface. As a result, according to the Twyman effect the gallium oxidesubstrate may be warped. In addition, when the second main surface ofthe gallium oxide substrate is detached from the upper surface plate andan entire surface is adsorbed to a flat chuck surface, the first mainsurface is deformed in the same shape as that of the lower surface ofthe upper surface plate. Thus, the irregularities of the lower surfaceof the upper surface plate may appear on the first main surface.

Conventionally, flatness of gallium oxide substrates has been poor, andthe transfer accuracy of exposure patterns to the gallium oxidesubstrates has been low.

An aspect of the present disclosure provides a technique that canimprove a flatness of a gallium oxide substrate and can accuratelytransfer an exposure pattern to the gallium oxide substrate.

Means for Solving Problems

According to an aspect of the present disclosure, a gallium oxidesubstrate includes a first main surface; and a second main surface whichis opposite to the first main surface.

When measured data z₀(r, θ) of height differences of points (r, θ, z) onthe first main surface from a reference plane, which is a least squareplane of the first main surface, are approximated by a function z(r, θ)expressed by equation (1), j is an index presenting a combination of nand k, expressed by equation (4), a_(nm) is a coefficient obtained byequation (5), parameters (r, θ) are polar coordinates on the referenceplane, n is an integer greater than or equal to 0 and less than or equalto k, k is 16, m is an even number within a range from −n to +n when nis an even number, and m is an odd number within a range from −n to +nwhen n is an odd number,

a ratio (PV1/D) of a first maximum height difference (PV1) of acomponent of z(r, θ) obtained by summing all terms a_(nm)z_(nm)(r, θ)with j which are 4, 9, 16, 25, 36, 49, 64, and 81, when the second mainsurface is placed facing a horizontal flat surface, to a diameter (D) ofthe first main surface is less than or equal to 0.39×10⁻⁴, and

a ratio (PV2/D) of a second maximum height difference (PV2) of acomponent of z(r, θ) obtained by summing all terms a_(nm)z_(nm)(r, θ)with j which are greater than or equal to 4 and less than or equal to81, when an entire surface of the second main surface is adsorbed to aflat chuck surface, to the diameter (D) of the first main surface isless than or equal to 0.59×10⁻⁴.

$\begin{matrix}{\left\lbrack {{Math}\mspace{14mu} 1} \right\rbrack\mspace{650mu}} & \; \\{{z\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{k}{\sum\limits_{m = {- n}}^{n}{a_{nm}{z_{nm}\left( {r,\theta} \right)}}}}} & (1) \\{\left\lbrack {{Math}\mspace{14mu} 2} \right\rbrack\mspace{650mu}} & \; \\{{z_{nm}\left( {r,\theta} \right)} = \left\{ \begin{matrix}{{R_{n}^{m}(r)}{\cos\left( {m\;\theta} \right)}} & {m \geq 0} \\{{R_{n}^{m}(r)}{\sin\left( {{m}\theta} \right)}} & {m < 0}\end{matrix} \right.} & (2) \\{\left\lbrack {{Math}\mspace{14mu} 3} \right\rbrack\mspace{650mu}} & \; \\{{R_{n}^{m}(r)} = \left\{ \begin{matrix}{\sum\limits_{i = 0}^{\frac{n - m}{2}}{\frac{\left( {- 1} \right)^{i}{\left( {n - i} \right)!}}{i{!{\left( {\frac{n + m}{2} - i} \right){!{\left( {\frac{n - m}{2} - i} \right)!}}}}}r^{n - {2i}}}} & {{n - {m\mspace{14mu}{is}\mspace{14mu}{even}}}\ } \\0 & {n - {m\mspace{14mu}{is}\mspace{14mu}{odd}}}\end{matrix} \right.} & (3) \\{\left\lbrack {{Math}\mspace{14mu} 4} \right\rbrack\mspace{650mu}} & \; \\{j = {\left( {1 + \frac{n + {m}}{2}} \right)^{2} - {2{m}} + \left\lfloor \frac{1 - {{sgn}\; m}}{2} \right\rfloor}} & (4) \\{\left\lbrack {{Math}\mspace{14mu} 5} \right\rbrack\mspace{650mu}} & \; \\{a_{n\; m} = \frac{\int_{0}^{2\pi}{\int_{0}^{D/2}{{z_{0}\left( {r,\theta} \right)}{z_{nm}\left( {r,\theta} \right)}rdrd\theta}}}{\int_{0}^{2\pi}{\int_{0}^{D/2}{{z_{nm}\left( {r,\theta} \right)}{z_{nm}\left( {r,\theta} \right)}rdrd\theta}}}} & (5)\end{matrix}$

Effects of the Invention

According to the aspect of the present disclosure, a flatness of agallium oxide substrate can be improved, and an exposure pattern can betransferred to the gallium oxide substrate with high accuracy.

BRIEF DESCRIPTION OF DRAWINGS

Other objects and further features of the present disclosure will beapparent from the following detailed description when read inconjunction with the accompanying drawings, in which:

FIG. 1 is a flowchart illustrating a method of manufacturing a galliumoxide substrate according to an embodiment of the present disclosure;

FIG. 2 is a perspective view illustrating an example of a single-sidedpolishing device for performing the first stage single-sided polishingshown in FIG. 1;

FIG. 3 is a cross-sectional view illustrating the example of thesingle-sided polishing device for performing the first stagesingle-sided polishing in FIG. 1;

FIG. 4 is a perspective view illustrating an example of a double-sidedpolishing device for performing the double-sided polishing shown in FIG.1;

FIG. 5 is a cross-sectional view illustrating the example of thedouble-sided polishing device for performing the double-sided polishingshown in FIG. 1;

FIG. 6 is a cross-sectional view illustrating an example of the galliumoxide substrate when a first maximum height difference (PV1) ismeasured;

FIG. 7 is a diagram showing z_(nm)(r, θ) for j=1 (n=0, m=0), j=2 (n=1,m=1), j=4 (n=2, m=0), and j=9 (n=4, m=0), respectively; and

FIG. 8 is a cross-sectional view illustrating an example of the galliumoxide substrate when a second maximum height difference (PV2) ismeasured.

MODE FOR CARRYING OUT THE INVENTION

In the following, embodiments of the present disclosure will bedescribed with reference to the drawings. In crystallographicdescriptions in the specification of the present disclosure, individualorientations are indicated by [ ], collective orientations are indicatedby < >, individual planes are indicated by ( ), and collective planesare indicated by { }. A negative crystallographic exponent is usuallyrepresented by a bar above a numeral, but in the specification of thepresent application the negative crystallographic exponent will berepresented by a negative sign before the numeral.

FIG. 1 is a flowchart illustrating a method of manufacturing a galliumoxide substrate according to an embodiment of the present disclosure. Asshown in FIG. 1, the method of manufacturing the gallium oxide substrateincludes a first stage single-sided polishing of the gallium oxidesubstrate (Step S1). For the gallium oxide substrate, for example, aβ-Ga₂O₃ single crystal preliminarily sliced into a plate using a wiresaw or the like and ground to a predetermined thickness using a grindingdevice or the like, is used. The gallium oxide substrate may includedopants or may not include dopants. Suitable dopants may include, forexample, Si, Sn, Al or In.

FIG. 2 is a perspective view illustrating an example of a single-sidedpolishing device for performing the first stage single-sided polishingshown in FIG. 1. FIG. 3 is a cross-sectional view illustrating theexample of the single-sided polishing device for performing the firststage single-sided polishing shown in FIG. 1. In FIG. 3, irregularitiesof a lower surface 121 of an upper surface plate 120 are exaggerated. Asingle-sided polishing device for performing the second stagesingle-sided polishing (step S2) shown in FIG. 1 is the same as thesingle-sided polishing device 100 shown in FIG. 2 and FIG. 3, and is notshown.

The single-sided polishing device 100 includes a lower surface plate110, the upper surface plate 120, and a nozzle 130. The lower surfaceplate 110 is arranged horizontally, and a lower polishing pad 112 isattached to an upper surface 111 of the lower surface plate 110. Theupper surface plate 120 is arranged horizontally, and the gallium oxidesubstrate 10 is fixed to a lower surface 121 of the upper surface plate120. The upper surface plate 120 holds the gallium oxide substrate 10horizontally, and presses the gallium oxide substrate 10 against thelower polishing pad 112. The lower polishing pad 112 may be absent, inwhich case the upper surface plate 120 presses the gallium oxidesubstrate 10 against the lower surface plate 110. A diameter of theupper surface plate 120 is less than a radius of the lower surface plate110, and the upper surface plate 120 is disposed radially outward of arotational center line C1 of the lower surface plate 110. The rotationalcenter line C2 of the upper surface plate 120 is parallel to therotational center line C1 of the lower surface plate 110. The lowersurface plate 110 is rotated around the center line C1. The uppersurface plate 120 is rotated passively with the rotation of the lowersurface plate 110. The upper surface plate 120 may be rotatedindependently of the lower surface plate 110, or may be rotated by adifferent motor.

The gallium oxide substrate 10 has a first main surface 11 with acircular shape and a second main surface 12 with a circular shapeopposite to the first main surface 11. On an outer periphery of thegallium oxide substrate 10, a notch or the like which is not shown toindicate a crystal orientation of the gallium oxide is formed. Anorientation flat may be formed instead of the notch. The first mainsurface 11 is, for example, a {001} plane. The {001} plane is a crystalplane perpendicular to the <001> direction, and may be either a (001)plane or a (00−1) plane.

In addition, the first main surface 11 may be a crystal plane other thanthe {001} plane. Moreover, the first main surface 11 may also have anoff angle with respect to a predetermined crystal plane. The off angleimproves crystallinity of an epitaxial film formed on the first mainsurface 11 after the polishing.

The nozzle 130 supplies a polishing slurry 140 to the lower polishingpad 112. The polishing slurry 140 includes, for example, particles andwater. In this case, the particles are dispersoids and the water is adispersion medium. The dispersion medium may be an organic solvent. Thepolishing slurry 140 is supplied between the gallium oxide substrate 10and the lower polishing pad 112, and used for polishing the lowersurface of the gallium oxide substrate 10 to be flat.

In the first stage single-sided polishing (step S1), for example,diamond particles are used for the particles. The Mohs hardness ofdiamond particles is 10. A median diameter D50 of the diamond particlesis not particularly limited, and is, for example, 50 μm. The mediandiameter “D50” represents a 50% diameter in volume based cumulativefractions of a particle diameter distribution measured by a dynamiclight scattering method. The dynamic light scattering method is a methodfor measuring particle diameter distribution by irradiating thepolishing slurry 140 with laser light and observing scattered light witha photodetector.

In the first stage single-sided polishing (step S1), the first mainsurface 11 of the gallium oxide substrate 10 is pressed against thelower polishing pad 112 and polished to be flat by the lower polishingpad 112 and the polishing slurry 140. The second main surface 12 of thegallium oxide substrate 10 is fixed to the lower surface 121 of theupper surface plate 120, and irregularities of the lower surface 121 aretransferred to the second main surface 12.

The upper surface 111 of the lower surface plate 110 also hasirregularities in the same manner as the lower surface 121 of the uppersurface plate 120, but the irregularities are unlikely to be transferredto the first main surface 11 of the gallium oxide substrate 10.Different from the upper surface plate 120, the lower surface plate 110is displaced relative to the gallium oxide substrate 10.

As shown in FIG. 1, the method of manufacturing a gallium oxidesubstrate includes a second stage single-sided polishing of the galliumoxide substrate (step S2). In second stage single-sided polishing (stepS2), in the same manner as the first stage single-sided polishing (stepS1), the first main surface 11 of the gallium oxide substrate is pressedagainst the lower polishing pad 112, and polished to be flat by thelower polishing pad 112 and the polishing slurry 140.

In the second stage single-sided polishing (step S2), particles with asmaller median diameter D50 and lower Mohs hardness (i.e. softer) thanthose of the first stage single-sided polishing (step S1) may be used.For example, colloidal silica may be used for the particles. The secondmain surface 12 of the gallium oxide substrate 10 is fixed to the lowersurface 121 of the upper surface plate 120, and the irregularities ofthe lower surface 121 are transferred to the second main surface 12.

As described above, the upper surface 111 of the lower surface plate 110also has irregularities in the same manner as the lower surface 121 ofthe upper surface plate 120, but the irregularities are unlikely to betransferred to the first main surface 11 of the gallium oxide substrate10. Different from the upper surface plate 120, the lower surface plate110 is displaced relative to the gallium oxide substrate 10.

In the first stage single-sided polishing (step S1) and the second stagesingle-sided polishing (step S2), only one side (the first main surface11) is polished. Then, a residual stress of the first main surface 11after the polishing becomes different from a residual stress of thesecond main surface 12. As a result, the gallium oxide substrate 10 maybe warped according to the Twyman effect. Moreover, when the second mainsurface 12 of the gallium oxide substrate 10 is detached from the uppersurface plate 120, and the entire surface is adsorbed to a flat chucksurface, the first main surface 11 is deformed in the same shape as thatof the lower surface 121 of the upper surface plate 120. Thus, theirregularities of the lower surface 121 of the upper surface plate 120may appear on the first main surface 11.

Thus, as shown in FIG. 1, the method of manufacturing the gallium oxidesubstrate further includes polishing the gallium oxide substrate on bothsides (step S3). Different from the first stage single-sided polishing(step S1) and the second stage single-sided polishing (step S2), thedouble-sided polishing (step S3) includes polishing the first mainsurface 11 and the second main surface 12 simultaneously.

FIG. 4 is a perspective view illustrating an example of a double-sidedpolishing device for performing the double-sided polishing shown inFIG. 1. FIG. 5 is a cross-sectional view illustrating the example of thedouble-sided polishing device for performing the double-sided polishingshown in FIG. 1. The double-sided polishing device 200 includes a lowersurface plate 210, an upper surface plate 220, a carrier 230, a sun gear240, and an internal gear 250. The lower surface plate 210 is arrangedhorizontally and a lower polishing pad 212 is attached to an uppersurface 211 of the lower surface plate 210. The upper surface plate 220is arranged horizontally, and an upper polishing pad 222 is applied to alower surface 221 of the upper surface plate 220. The carrier 230 holdsthe gallium oxide substrate 10 horizontally between the lower surfaceplate 210 and the upper surface plate 220. The carrier 230 is disposedradially outward of the sun gear 240 and radially inward of the internalgear 250. The sun gear 240 and the internal gear 250 are arrangedconcentrically and are engaged with an outer peripheral gear 231 of thecarrier 230.

The double-sided polishing device 200 is, for example, a four-waydouble-sided polishing device in which the lower surface plate 210, theupper surface plate 220, the sun gear 240, and the internal gear 250rotate about the same vertical rotational center line. The lower surfaceplate 210 and the upper surface plate 220 rotate in opposite directionsto each other, and press the lower polishing pad 212 against the lowersurface of the gallium oxide substrate 10 and press the upper polishingpad 222 against the upper surface of the gallium oxide substrate 10. Atleast one of the lower surface plate 210 and the upper surface plate 220supply a polishing slurry to the gallium oxide substrate 10. Thepolishing slurry is supplied between the gallium oxide substrate 10 andthe lower polishing pad 212, and used for polishing the lower surface ofthe gallium oxide substrate 10. Moreover, the polishing slurry is alsosupplied between the gallium oxide substrate 10 and the upper polishingpad 222, and used for polishing the upper surface of the gallium oxidesubstrate 10.

For example, the lower surface plate 210, the sun gear 240, and theinternal gear 250 rotate in the same direction in a top view. Theserotation directions are opposite to the rotation direction of the uppersurface plate 220. The carrier 230 revolves around the rotational centerline while turning on its axis. The revolving direction of the carrier230 is the same direction as the rotation direction of the sun gear 240and the internal gear 250. The turning direction of the carrier 230 onits axis is determined by whether a product of a rotation speed and apitch circle diameter of the sun gear 240 is greater than a product of arotation speed and a pitch circle diameter of the internal gear 250. Ifthe product of the rotation speed and the pitch circle diameter of theinternal gear 250 is greater than the product of the rotation speed andthe pitch circle diameter of the sun gear 240, the turning direction ofthe carrier 230 on its axis is the same direction as the revolvingdirection of the carrier 230 around the rotational center line. If theproduct of the rotation speed and the pitch circle diameter of theinternal gear 250 is less than the product of the rotation speed and thepitch circle diameter of the sun gear 240, the turning direction of thecarrier 230 on its axis is opposite to the revolving direction of thecarrier 230 around the rotational center line.

The double-sided polishing device 200 may be a three-way double-sidedpolishing device or a two-way double-sided polishing device. Thethree-way double-sided polishing device may be any of, for example, (1)a double-sided polishing device in which the internal gear is fixed, andthe lower surface plate 210, the upper surface plate 220, and the sungear are rotated and (2) a double-sided polishing device in which theupper surface plate 220 is fixed, and the lower surface plate 210, thesun gear 240, and the internal gear 250 are rotated. Moreover, thetwo-way double-sided polishing device is, for example, a device in whichthe lower surface plate 210 and the upper surface plate 220 are fixed,and the sun gear 240 and the internal gear 250 are rotated.

The carrier 230 holds the gallium oxide substrate 10 horizontally, forexample, with the first main surface 11 of the gallium oxide substratefacing down. The carrier 230 may hold the gallium oxide substrate 10horizontally with the first main surface 11 of the gallium oxidesubstrate facing up. In either case, the first main surface 11 and thesecond main surface 12 of the gallium oxide substrate 10 are polishedsimultaneously.

Because in the double-sided polishing (step S3), different from thefirst stage single-sided polishing (step S1) and the second stagesingle-sided polishing (step S2), the first main surface 11 and thesecond main surface 12 are polished simultaneously, the differencebetween the residual stress of the first main surface 11 and theresidual stress of the second main surface 12 after the polishing can bereduced. Thus, the warpage due to the Twyman effect can be reduced.

The warpage due to the Twyman effect will be assessed by using a firstmaximum height difference (PV1) which will be described later. FIG. 6 isa diagram depicting a side view of the gallium oxide substrate when thefirst maximum height difference (PV1) is measured. As shown in FIG. 6,when the first maximum height difference (PV1) is measured, the galliumoxide substrate is placed with the second main surface 12 facing ahorizontal flat surface 20 so that the gallium oxide substrate 10 is notdeformed. In FIG. 6, an xy-plane including an x-axis and a y-axisorthogonal to each other are a least square plane of the first mainsurface 11. The least square plane of the first main surface 11 is aplane obtained by approximating the first main surface 11 by the leastsquares method. Moreover, in FIG. 6, a z-axis orthogonal to the x-axisand the y-axis is set to pass through a center of the first main surface11.

Measured data z₀(r, θ) of the height difference of the first mainsurface 11 from the least square plane of the first main surface 11, asa reference plane 13, are approximated by z(r, θ) of the followingequation (1).

$\begin{matrix}{\left\lbrack {{Math}\mspace{14mu} 6} \right\rbrack\mspace{661mu}} & \; \\{{z\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{k}{\sum\limits_{m = {- n}}^{n}{a_{nm}{z_{nm}\left( {r,\theta} \right)}}}}} & (1) \\{\left\lbrack {{Math}\mspace{14mu} 7} \right\rbrack\mspace{661mu}} & \; \\{{z_{nm}\left( {r,\theta} \right)} = \left\{ \begin{matrix}{{R_{n}^{m}(r)}{\cos\left( {m\;\theta} \right)}} & {m \geq 0} \\{{R_{n}^{m}(r)}{\sin\left( {{m}\theta} \right)}} & {m < 0}\end{matrix} \right.} & (2) \\{\left\lbrack {{Math}\mspace{14mu} 8} \right\rbrack\mspace{661mu}} & \; \\{{R_{n}^{m}(r)} = \left\{ \begin{matrix}{\sum\limits_{i = 0}^{\frac{n - m}{2}}{\frac{\left( {- 1} \right)^{i}{\left( {n - i} \right)!}}{i{!{\left( {\frac{n + m}{2} - i} \right){!{\left( {\frac{n - m}{2} - i} \right)!}}}}}r^{n - {2i}}}} & {{n - {m\mspace{14mu}{is}\mspace{14mu}{even}}}\ } \\0 & {n - {m\mspace{14mu}{is}\mspace{14mu}{odd}}}\end{matrix} \right.} & (3) \\{\left\lbrack {{Math}\mspace{14mu} 9} \right\rbrack\mspace{661mu}} & \; \\{j = {\left( {1 + \frac{n + {m}}{2}} \right)^{2} - {2{m}} + \left\lfloor \frac{1 - {{sgn}\; m}}{2} \right\rfloor}} & (4) \\{\left\lbrack {{Math}\mspace{14mu} 10} \right\rbrack\mspace{644mu}} & \; \\{a_{n\; m} = \frac{\int_{0}^{2\pi}{\int_{0}^{D/2}{{z_{0}\left( {r,\theta} \right)}{z_{nm}\left( {r,\theta} \right)}rdrd\theta}}}{\int_{0}^{2\pi}{\int_{0}^{D/2}{{z_{nm}\left( {r,\theta} \right)}{z_{nm}\left( {r,\theta} \right)}rdrd\theta}}}} & (5)\end{matrix}$

where in the equations (1) to (5), (r, θ) are polar coordinates on thereference plane 13, n is a natural number greater than or equal to 0 andless than or equal to k, k is 16, m is even numbers within a range from−n to +n when n is an even number, m is odd numbers within a range from−n to +n when n is an odd number, j is an index representing acombination of n and k, and a_(nm) is a coefficient. As shown in theequation (4), in the embodiment, the Fringe notation is used forexpressing a combination of two indices n and m by a single index j. Theequation (2) expresses a Zernike polynomial. Because the Zernikepolynomials are orthogonal polynomials, the coefficients a_(nm) can beobtained by the equation (5).

FIG. 7 is a diagram showing z_(nm)(r, θ) with j=1 (n=0, m=0), j=2 (n=1,m=1), j=4 (n=2, m=0), and j=9 (n=4, m=0), respectively.

As shown by a solid line in FIG. 7, z_(nm)(r, θ) with j=1 is an offsetplane parallel to the xy-plane. The z_(nm)(r, θ) with j=1 is independentof r and θ.

As shown by a dashed line in FIG. 7, z_(nm)(r, θ) with j=2 is aninclined plane obtained by the xy-plane around the y-axis. Moreover,z_(nm)(r, θ) with j=3 (n=1, m=−1) is an inclined plane obtained byrotating the xy-plane around the x-axis.

As shown by a dotted chain line in FIG. 7, z_(nm)(r, θ) with j=4 is acurved surface obtained by rotating a quadratic curve on the xz-planesymmetric with respect to the z-axis by 180 degrees around the z-axis.The z_(nm)(r, θ) with j=4 depends on r, and is independent of θ.

As shown by a two-dot chain line in FIG. 7, z_(nm)(r, θ) with j=9 is acurved surface obtained by rotating a quartic curve on the xz-planesymmetric with respect to the z-axis by 180 degrees around the z-axis.The z_(nm)(r, θ) with j=9 depends on r, and is independent of θ.

When j is a square of a natural number (e.g. 4, 9, 16, 25, 36, 49, 64,81, or the like), the z_(nm) (r, θ) depends on r, and is independent ofθ. In addition, the z_(nm)(r, θ) with j=1 (n=0, m=0) is independent ofeither r or θ as described above.

The warpage due to the Twyman effect is caused by the difference betweenthe residual stress of the first main surface 11 and the residual stressof the second main surface 12. The residual stress difference depends onr and is independent of θ.

Thus, the warpage due to the Twyman effect will be evaluated by thefirst maximum height difference (PV1) of the component of z(r, θ)obtained by summing all terms a_(nm)z_(nm)(r, θ) with j which are 4, 9,16, 25, 36, 49, 64, and 81. The first maximum height difference (PV1) isa height difference between the highest point with respect to thereference plane 13 and the lowest point with respect to the referenceplane 13. The smaller the warpage due to the Twyman effect is, thesmaller the first maximum height difference (PV1) is.

In addition, the terms a_(nm)z_(nm)(r, θ) with j which is greater than81 will be ignored because these terms have almost no effect onirregularities of the first main surface 11. Thus, the calculationbecomes simpler.

In the double-sided polishing (step S3), different from the first stagesingle-sided polishing (step S1) and the second stage single-sidedpolishing (step S2), the first main surface 11 and the second mainsurface 12 are polished simultaneously, so that the warpage due to theTwyman effect can be reduced, as described above. As a result, a ratio(PV1/D) of the first maximum height difference (PV1) to the diameter (D)of the first main surface 11 is reduced to 0.39×10⁻⁴ or less. Inaddition, the first maximum height difference (PV1) can be reduced to 2μm or less. In addition, the ratio PV1/D is a dimensionless quantity,and “10⁻⁴” in the value of the ratio PV1/D can be regarded to beequivalent to “μm/cm”.

The ratio PV1/D is, for example, less than or equal to 0.39×10⁻⁴ asdescribed above. When the ratio PV1/D is less than or equal to0.39×10⁻⁴, the warpage due to the Twyman effect can be reduced, so thatthe flatness of the gallium oxide substrate 10 can be improved, andconsequently, an exposure pattern can be transferred to the galliumoxide substrate 10 with high accuracy. The ratio PV1/D is preferably0.2×10⁻⁴ or less, and more preferably 0.1×10⁻⁴ or less. Moreover, thePV1/D is preferably 0.02×10⁻⁴ or more from a viewpoint of productivity.

The first maximum height difference PV1 is 2 μm or less, for example, asdescribed above. When the first maximum height difference PV1 is 2 μm orless, the warpage due to the Twyman effect can be reduced, so that theflatness of the gallium oxide substrate 10 can be improved, andconsequently, an exposure pattern can be transferred to the galliumoxide substrate 10 with high accuracy. The first maximum heightdifference PV1 is preferably 1 μm or less, and more preferably 0.5 μm orless. The first maximum height difference PV1 is preferably 0.1 μm ormore from the viewpoint of productivity.

The diameter D of the first main surface 11 is not particularly limited,but is, for example, within a range from 5 cm to 31 cm, preferablywithin a range from 10 cm to 21 cm, and more preferably within a rangefrom 12 cm to 15 cm.

In the double-sided polishing (step S3), different from the first stagesingle-sided polishing (step S1) and the second stage single-sidedpolishing (step S2), the lower surface plate 210 and the upper surfaceplate 220 are displaced relative to the gallium oxide substrate 10. As aresult, transfer of the irregularities of the lower surface 221 of theupper surface plate 220 to the upper surface of the gallium oxidesubstrate 10 is suppressed, and the upper surface of the gallium oxidesubstrate 10 can be polished so as to be parallel to the lower surfaceof the gallium oxide substrate 10. Accordingly, when an entire surfaceof the second main surface 12 of the gallium oxide substrate 10 isadsorbed to a flat chuck surface 30, it is possible to prevent theirregularities of the lower surface 221 of the upper surface plate 220from appearing on the first main surface 11.

The shape transfer of the upper surface plate 220 to the gallium oxidesubstrate 10 is evaluated by a second maximum height difference (PV2).FIG. 8 is a side view of the gallium oxide substrate when the secondmaximum height difference (PV2) is measured. As shown in FIG. 8, thesecond maximum height difference (PV2) is measured in a state where anentire surface of the second main surface 12 is adsorbed to the flatchuck surface 30. The adsorption is, for example, vacuum adsorption, andthe chuck surface 30 is formed of a porous material. In FIG. 8, thexy-plane including the x-axis and the y-axis orthogonal to each other isthe least square plane of the first main surface 11. Moreover, in FIG.8, the z-axis orthogonal to the x-axis and the y-axis is set to passthrough the center of the first main surface 11.

The measured data z₀(r, θ) of the height difference of the first mainsurface 11 from the reference plane 13, which is the least square planeof the first main surface 11, is approximated by z(r, θ) of theabove-described equation (1). The z_(nm)(r, θ) of j=1, 2, and 3represents flat planes as described above, and is not a relativecomponent when measuring the second maximum height difference (PV2).

Thus, the shape transfer of the upper surface plate 220 to the galliumoxide substrate 10 is evaluated by the second maximum height difference(PV2) of the component of z(r, θ) obtained by adding all a_(nm)z_(nm)(r,θ) with j which are greater than or equal to 4 and less than or equal to81. The second maximum height difference (PV2) is a difference betweenthe highest point with respect to the reference plane 13 and the lowestpoint with respect to the reference plane 13. The smaller the shapetransfer of the upper surface plate 220 to the gallium oxide substrate10 is, the smaller the second maximum height difference (PV2) is.

Note that the terms a_(nm)z_(nm)(r, θ) with j which is greater than 81do not contribute to the irregularities of the first main surface 11,and thus the terms will be neglected for simplicity.

In the double-sided polishing (step S3), different from the first stagesingle-sided polishing (step S1) and the second stage single-sidedpolishing (step S2), the first main surface 11 and the second mainsurface 12 are polished simultaneously, so that the shape transfer ofthe upper surface plate 220 to the gallium oxide substrate 10 can besuppressed, as described above. As a result, the ratio (PV2/D) of thesecond maximum height difference (PV2) to the diameter (D) of the firstmain surface 11 can be reduced to 0.59×10⁻⁴ or less. In addition, thesecond maximum height difference (PV2) can be reduced to 3 μm or less.In addition, the ratio PV2/D is a dimensionless quantity, and “10⁻⁴” inthe value of the ratio PV2/D can be regarded to be equivalent to“μm/cm”.

The ratio PV2/D is, for example, less than 0.59×10⁻⁴, as describedabove. If the ratio PV2/D is less than or equal to 0.59×10⁻⁴, the shapetransfer of the upper surface plate 220 to the gallium oxide substrate10 can be suppressed. Thus, the flatness of the gallium oxide substrate10 can be improved, and consequently, the exposure pattern can betransferred to the gallium oxide substrate 10 with high accuracy. Theratio PV2/D is preferably 0.2×10⁻⁴ or less, and more preferably 0.1×10⁻⁴or less. Moreover, the ratio PV2/D is preferably 0.02×10⁻⁴ or more fromthe viewpoint of productivity.

The second maximum height difference PV2 is, for example, 3 μm or less,as described above. If the second maximum height difference PV2 is 3 μmor less, the shape transfer of the upper surface plate 220 to thegallium oxide substrate 10 can be suppressed, so that the flatness ofthe gallium oxide substrate 10 can be improved, and consequently, theexposure pattern can be transferred to the gallium oxide substrate 10with high accuracy. The second maximum height difference PV2 ispreferably 1 μm or less, and more preferably 0.5 μm or less. The secondmaximum height difference PV2 is preferably 0.1 μm or more from theviewpoint of productivity.

The double-sided polishing (step S3) includes polishing the first mainsurface 11 and the second main surface 12 of the gallium oxide substrate10 simultaneously, in opposite directions to each other, with apolishing slurry containing particles having a Mohs hardness of 7 orless. If the Mohs hardness is 7 or less, the particles are soft, so thatan occurrence of scratch on a surface of the gallium oxide substrate 10can be suppressed, and cracking of the gallium oxide substrate 10 can besuppressed. The Mohs hardness is preferably 6 or less, and morepreferably 5 or less. The Mohs hardness is preferably 2 or more from theviewpoint of the polishing speed.

For example, for the particle having a Mohs hardness of 7 or less,colloidal silica is used. The Mohs hardness of colloidal silica is 7.The material of the particles having the Mohs hardness of 7 or less isnot limited to SiO₂. The material may be TiO₂, ZrO₂, Fe₂O₃, ZnO, orMnO₂. The Mohs hardness of TiO₂ is 6, the Mohs hardness of ZrO₂ is 6.5,the Mohs hardness of Fe₂O₃ is 6, the Mohs hardness of ZnO is 4.5, andthe Mohs hardness of MnO₂ is 3. The polishing slurry used in thedouble-sided polishing (step S3) is required not to contain particleshaving the Mohs hardness greater than 7, and may contain two or moretypes of particles having the Mohs hardness of 7 or less.

In the double-sided polishing (step S3), the median diameter D50 of theparticles contained in the polishing slurry is, for example, 1 μm orless. If the median diameter D50 is 1 μm or less, the particles aresmall, so that an excessive stress on the gallium oxide substrate 10 canbe suppressed, and cracking of the gallium oxide substrate 10 can besuppressed. The median diameter D50 is preferably 0.7 μm or less, andmore preferably 0.5 μm or less. The median diameter D50 is preferably0.01 μm or more from the viewpoint of the polishing speed.

In the first half of the double-sided polishing (step S3), for example,polishing pressure is 9.8 kPa or less. In the first half of thedouble-sided polishing (step S3), since the first main surface 11 andthe second main surface 12 are not sufficiently flat, the irregularitiesare large, and stress concentration easily occurs. When the polishingpressure is 9.8 kPa or less during a period of 50% or more of the firsthalf of the double-sided polishing (step S3), an excessive stress on thegallium oxide substrate 10 is suppressed, and thereby cracking of thegallium oxide substrate 10 is suppressed. During the period of 50% ormore of the first half of the double-sided polishing (step S3), thepolishing pressure is preferably 8.8 kPa or less, and more preferably7.8 kPa or less. In addition, from the viewpoint of the polishing speed,the polishing pressure is preferably 3 kPa or more during the period of50% or more of the first half of the double-sided polishing (step S3).

During the entire period of the double-sided polishing (step S3), thepolishing pressure may be constant. In the double-sided polishing (S3),the first main surface 11 and the second main surface 12 are graduallyplanarized, and the irregularities become gradually smaller. Therefore,the polishing pressure may be increased in order to improve thepolishing speed.

The method of manufacturing the gallium oxide substrate is not limitedto that shown in FIG. 1, and may be a method that includes thedouble-sided polishing (step S3). The method of manufacturing thegallium oxide substrate may include a process other than the processesshown in FIG. 1, for example, it may include a cleaning process offlushing off deposits (e.g. particles) of the gallium oxide substrate10. The cleaning process is performed, for example, between the firststage single-sided polishing (step S1) and the second stage single-sidedpolishing (step S2) and between the second stage single-sided polishing(step S2) and the double-sided polishing (step S3).

EXAMPLE

Hereinafter, examples and comparative examples will be described.Examples 1 to 3 are practical examples and Examples 4 to 7 arecomparative examples.

Examples 1 to 3

In Examples 1 to 3, the first stage single-sided polishing (step S1),the second stage single-sided polishing (step S2), and the double-sidedpolishing (step S3) were performed for a β-Ga₂O₃ single crystalsubstrate having a diameter of 50.8 mm and a thickness of 0.7 mm underthe same condition as shown in FIG. 1.

In the first stage single-sided polishing (step S1), a (001) surface ofthe β-Ga₂O₃ single-crystal substrate was polished by the single-sidedpolishing device 100 shown in FIG. 2. A lower surface plate 110 made oftin and diamond particles having a particle diameter of 0.5 μm was used.In the first stage single-sided polishing (step S1), the substrate ispressed against the lower surface plate 110 and polished without usingthe lower polishing pad 112.

In the second stage single-sided polishing (step S2), the (001) surfaceof the β-Ga₂O₃ single-crystal substrate was polished by the single-sidedpolishing device 100 shown in FIG. 2. In the second stage single-sidedpolishing (step S2), different from the first stage single-sidedpolishing (step S1), the lower polishing pad 112 was used. In the secondstage single-sided polishing (step S2), a lower polishing pad 112 madeof polyurethane and colloidal silica particles having a particlediameter of 0.05 μm was used.

In the double-sided polishing (step S3), the (001) and (00−1) surfacesof the β-Ga₂O₃ single crystal substrate were simultaneously polished bythe double-sided polishing device 200 shown in FIG. 4. For thedouble-sided polishing device 200, DSM9B by SpeedFam Co., Ltd. was used.For the lower polishing pad 212 and the upper polishing pad 222, N7512by FILWEL Co., Ltd. was used. The polishing slurry contained 20% by massof colloidal silica and 80% by mass of water. The median diameter D50 ofthe colloidal silica was 0.05 μm. During the entire period of thedouble-sided polishing (step S3), the polishing pressure was 9.8 kPa.The rotation speed of the lower surface plate 210 was 40 rpm, therotation speed of the upper surface plate 220 was 14 rpm, the rotationspeed of the sun gear 240 was 9 rpm, and the rotation speed of theinternal gear 250 was 15 rpm. The pitch circle diameter of the sun gear240 was 207.4 mm, and the pitch circle diameter of the internal gear 250was 664.6 mm.

Examples 4 to 6

In Examples 4 to 6, for a β-Ga₂O₃ single crystal substrate having adiameter of 50.8 mm and a thickness of 0.7 mm, only the first stagesingle-sided polishing (step S1) and the second stage single-sidedpolishing (step S2) were performed under the same condition as inExamples 1 to 3. That is, in Examples 4 to 6, the double-sided polishing(step S3) was not performed.

Example 7

In Example 7, the first stage single-sided polishing (step S1), thesecond stage single-sided polishing (step S2), and the double-sidedpolishing (step S3) were performed under the same conditions as inExamples 1 to 3, except that diamond particles having a particlediameter of 0.5 μm were used as the double-sided polishing (step S3)particles, and except that an epoxy resin was used as the polishing padfor the diamond particles. As a result, the gallium oxide substrate 10cracked during the double-sided polishing (step S3).

[Result of Polishing]

The first maximum height difference (PV1) of the (001) plane, which isthe first main surface 11, was measured in the state where the (00-1)plane, which is the second main surface 12, faces the horizontal flatsurface 20 so as not to deform the gallium oxide substrate 10, as shownin FIG. 6. For the measuring device, PF-60 by Mitaka Kohki Co., Ltd. wasused.

The second maximum height difference (PV2) of the (001) surface, whichis the first main surface 11, was measured in a state where an entiresurface of the (00−1) surface, which is the second main surface 12, isadsorbed to a flat chuck surface 30, as shown in FIG. 8. For themeasured device, PF-60 by Mitaka Kohki Co., Ltd. was used.

The polishing results of Examples 1 to 6 are shown in TABLE 1. Result ofExample 7 is not shown because the gallium oxide substrate 10 crackedduring the double-sided polishing (step S3) as described above.

TABLE 1 Polish- ing Double- pres- sided D50 sure PV1 PV2 polishing [μm][kPa] [μm] PV1/D [μm] PV2/D Ex. 1 Performed 0.05 9.8 0.8 0.16 × 10⁻⁴ 1.60.31 × 10⁻⁴ Ex. 2 Performed 0.05 9.8 0.6 0.12 × 10⁻⁴ 2.8 0.55 × 10⁻⁴ Ex.3 Performed 0.05 9.8 0.5 0.10 × 10⁻⁴ 1.2 0.24 × 10⁻⁴ Ex. 4 Not 0.05 9.83.6 0.71 × 10⁻⁴ 4.7 0.93 × 10⁻⁴ performed Ex. 5 Not 0.05 9.8 3.9 0.77 ×10⁻⁴ 5.9 1.16 × 10⁻⁴ performed Ex. 6 Not 0.05 9.8 4.5 0.89 × 10⁻⁴ 5.21.02 × 10⁻⁴ performed

As is obvious from TABLE 1, in Examples 1 to 3, different from Examples4 to 6, the gallium oxide substrate 10 was subjected to the double-sidedpolishing (step S3), so that the ratio PV1/D was less than or equal to0.39×10⁻⁴, and the first maximum height difference PV1 was 2 μm or less.The warpage due to the Twyman effect was found to be reduced byperforming the double-sided polishing (step S3).

Moreover, as is obvious from TABLE 1, since in Examples 1 to 3,different from Examples 4 to 6, the gallium oxide substrate 10 wassubjected to the double-sided polishing (step S3), the ratio PV2/D wasless than or equal to 0.59×10⁻⁴, and the second maximum heightdifference PV2 was 3 μm or less. The shape transfer of the upper surfaceplate 220 to the gallium oxide substrate 10 was found to be suppressedby performing the double-sided polishing (step S3).

Furthermore, in Examples 1 to 3, since the Mohs hardness of theparticles used in the double-sided polishing (step S3) was 7 or less,the median diameter D50 of the particles was 1 μm or less, and thepolishing pressure was 9.8 kPa or less during the period of 50% or moreof the first half of the double-sided polishing, the gallium oxidesubstrate did not crack during the double-sided polishing. On the otherhand, in Example 7, since the Mohs hardness of the particles used indouble-sided polishing (S3) exceeded 7, the gallium oxide substrate 10cracked during the double-sided polishing.

In the first stage single-sided polishing (step S1), the diamondparticles having the Mohs hardness of 10 were used for polishing, butthe gallium oxide substrate 10 did not break. In the single-sidedpolishing, the gallium oxide substrate is unlikely to crack comparedwith the double-sided polishing. Thus, the single-sided polishing isconsidered to be employed in Japanese Unexamined Patent ApplicationPublication No. 2016-13932.

As described above, preferred embodiments and practical examples of thepresent invention, with respect to a gallium oxide substrate and amethod of manufacturing the gallium oxide substrate, have been describedin detail. However, the present invention is not limited to theembodiment or the practical examples, but various variations,modification, replacements, additions, deletions and combinations may bemade without departing from the scope recited in claims.

What is claimed is:
 1. A gallium oxide substrate comprising: a firstmain surface; and a second main surface which is opposite to the firstmain surface, wherein when measured data z₀(r, θ) of height differencesof points (r, θ, z) on the first main surface from a reference plane,which is a least square plane of the first main surface, areapproximated by a function z(r, θ) expressed by equation (1),$\begin{matrix}{\left\lbrack {{Math}\mspace{14mu} 1} \right\rbrack\mspace{650mu}} & \; \\{{z\left( {r,\theta} \right)} = {\sum\limits_{n = 0}^{k}{\sum\limits_{m = {- n}}^{n}{a_{nm}{z_{nm}\left( {r,\theta} \right)}}}}} & (1) \\{where} & \; \\\begin{matrix}{\left\lbrack {{Math}\mspace{14mu} 2} \right\rbrack\mspace{650mu}} \\{{z_{nm}\left( {r,\theta} \right)} = \left\{ \begin{matrix}{{R_{n}^{m}(r)}{\cos\left( {m\;\theta} \right)}} & {m \geq 0} \\{{R_{n}^{m}(r)}{\sin\left( {{m}\theta} \right)}} & {m < 0}\end{matrix} \right.}\end{matrix} & (2) \\{and} & \; \\\begin{matrix}{\left\lbrack {{Math}\mspace{14mu} 3} \right\rbrack\mspace{650mu}} \\{{R_{n}^{m}(r)} = \left\{ \begin{matrix}{\sum\limits_{i = 0}^{\frac{n - m}{2}}{\frac{\left( {- 1} \right)^{i}{\left( {n - i} \right)!}}{i{!{\left( {\frac{n + m}{2} - i} \right){!{\left( {\frac{n - m}{2} - i} \right)!}}}}}r^{n - {2i}}}} & {{n - {m\mspace{14mu}{is}\mspace{14mu}{even}}}\ } \\0 & {n - {m\mspace{14mu}{is}\mspace{14mu}{odd}}}\end{matrix} \right.}\end{matrix} & (3)\end{matrix}$ j is an index representing a combination of n and k,expressed by equation (4), $\begin{matrix}{\left\lbrack {{Math}\mspace{14mu} 4} \right\rbrack\mspace{650mu}} & \; \\{j = {\left( {1 + \frac{n + {m}}{2}} \right)^{2} - {2{m}} + \left\lfloor \frac{1 - {{sgn}\; m}}{2} \right\rfloor}} & (4)\end{matrix}$ a_(nm) is a coefficient obtained by equation (5),$\begin{matrix}{\left\lbrack {{Math}\mspace{14mu} 5} \right\rbrack\mspace{650mu}} & \; \\{a_{n\; m} = \frac{\int_{0}^{2\pi}{\int_{0}^{D/2}{{z_{0}\left( {r,\theta} \right)}{z_{nm}\left( {r,\theta} \right)}rdrd\theta}}}{\int_{0}^{2\pi}{\int_{0}^{D/2}{{z_{nm}\left( {r,\theta} \right)}{z_{nm}\left( {r,\theta} \right)}rdrd\theta}}}} & (5)\end{matrix}$ parameters (r, θ) are polar coordinates on the referenceplane, n is an integer greater than or equal to 0 and less than or equalto k, k is 16, m is an even number within a range from −n to +n, when nis an even number, and m is an odd number within a range from −n to +n,when n is an odd number, a ratio (PV1/D) of a first maximum heightdifference (PV1) of a component of z(r, θ) obtained by summing all termsa_(nm)z_(nm)(r, θ) with j which are 4, 9, 16, 25, 36, 49, 64, and 81,when the second main surface is placed facing a horizontal flat surface,to a diameter (D) of the first main surface is less than or equal to0.39×10⁻⁴, and a ratio (PV2/D) of a second maximum height difference(PV2) of a component of z(r, θ) obtained by summing all termsa_(nm)z_(nm)(r, θ) with j which are greater than or equal to 4 and lessthan or equal to 81, when an entire surface of the second main surfaceis adsorbed to a flat chuck surface, to the diameter (D) of the firstmain surface is less than or equal to 0.59×10⁻⁴.
 2. The gallium oxidesubstrate according to claim 1, wherein the first maximum heightdifference (PV1) is 2 μm or less, and the second maximum heightdifference (PV2) is 3 μm or less.
 3. A method of manufacturing a galliumoxide substrate comprising: polishing a first main surface and a secondmain surface of the gallium oxide substrate simultaneously in oppositedirections to each other, with a polishing slurry containing particleshaving a Mohs hardness of 7 or less.
 4. The method of manufacturing agallium oxide substrate according to claim 3, wherein a 50% diameter involume based cumulative fractions of a particle diameter distributionmeasured by a dynamic light scattering method of the particles containedin the polishing slurry is 1 μm or less.
 5. The method of manufacturinga gallium oxide substrate according to claim 3, wherein a polishingpressure is 9.8 kPa or less during a period of 50% or more of a firsthalf of a period of polishing the first main surface and the second mainsurface simultaneously.